Options Theory

Options Theory is the mathematical framework for valuing derivative securities. At its core, it relies on the principle of no-arbitrage and the concept of risk-neutral valuation.

I. Foundational Concepts

Underlying Assets and Discounting

Options are derivatives, meaning their value is derived from an Underlying Asset ( $S$ ), typically a stock, index, or commodity. The Bond ( $B$ ) represents the risk-free rate ( $r$ ), used for discounting future cash flows.

Discount Factor: The present value of one unit of currency received at time $T$ is $e^{-rT}$ .
Vanilla Options:
- Call Option ( $C$ ): Right to buy the underlying at the Strike Price ( $K$ ) at time $T$ . Payoff: $\max(S_T - K, 0)$ .
- Put Option ( $P$ ): Right to sell the underlying at the Strike Price ( $K$ ) at time $T$ . Payoff: $\max(K - S_T, 0)$ .

Put-Call Parity

Put-Call Parity is a fundamental no-arbitrage relationship between the prices of a European call option, a European put option, the underlying stock, and a zero-coupon bond.

C + K e^{-rT} = P + S

This equation states that a portfolio consisting of a long call and a zero-coupon bond with face value $K$ (left side) must have the same value as a portfolio consisting of a long put and a long share of the stock (right side). Any deviation from this parity implies an arbitrage opportunity.

II. The Black-Scholes-Merton (BSM) Model

The BSM model provides a closed-form solution for pricing European options under several key assumptions, most notably that the underlying asset price follows a Geometric Brownian Motion (GBM).

The Black-Scholes Partial Differential Equation (PDE)

The BSM PDE is a second-order parabolic PDE that must be satisfied by the price of any derivative $V(S, t)$ that is a function of the underlying asset price $S$ and time $t$ , assuming no arbitrage.

\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} = rV

Interpretation: The equation represents the idea that a portfolio consisting of the derivative and a dynamically adjusted position in the underlying asset (the Delta-Hedge) must earn the risk-free rate $r$ .

The BSM Pricing Formula (European Call)

The solution to the PDE, with the call option payoff as the boundary condition, is:

C(S, t) = S N(d_1) - K e^{-r(T-t)} N(d_2)

where:

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}

d_2 = d_1 - \sigma \sqrt{T-t}

$N(\cdot)$ : Cumulative distribution function of the standard normal distribution.
Interpretation: $S N(d_1)$ is the expected present value of receiving the stock, and $K e^{-r(T-t)} N(d_2)$ is the expected present value of paying the strike price, both under the risk-neutral measure $\mathbb{Q}$ .

III. The Greeks: Risk Management and Hedging

The Greeks are the partial derivatives of the option price with respect to various input parameters. They are essential for understanding the sensitivity of an option's price and for constructing hedging strategies.

Greek	Formula (Partial Derivative)	Interpretation	Hedging Application
Delta ( $\Delta$ )	$\frac{\partial V}{\partial S}$	Change in option price for a one-unit change in the underlying price.	Primary Hedge: Used to create a delta-neutral portfolio (a portfolio whose value does not change with small movements in the underlying price).
Gamma ( $\Gamma$ )	$\frac{\partial^2 V}{\partial S^2}$	Change in Delta for a one-unit change in the underlying price.	Delta-Hedge Stability: Measures the effectiveness of the delta hedge. High Gamma means the hedge must be rebalanced frequently.
Theta ( $\Theta$ )	$\frac{\partial V}{\partial t}$	Change in option price for a one-unit change in time (time decay).	Time Risk: Measures the cost of holding the option over time. Typically negative for long options.
Vega ( $\mathcal{V}$ )	$\frac{\partial V}{\partial \sigma}$	Change in option price for a one-unit change in volatility ( $\sigma$ ).	Volatility Risk: Used to hedge against changes in the market's implied volatility.
Rho ( $\rho$ )	$\frac{\partial V}{\partial r}$	Change in option price for a one-unit change in the risk-free rate ( $r$ ).	Interest Rate Risk: Less critical than other Greeks but relevant for long-dated options.

IV. Advanced Concepts

Implied Volatility and the Volatility Smile

Implied Volatility ( $\sigma_{implied}$ ): The value of $\sigma$ that, when plugged into the BSM formula, yields the current market price of the option. It is a forward-looking measure of the market's expectation of future volatility.
Volatility Smile/Skew: The empirical observation that implied volatility is not constant across different strike prices and maturities, contradicting the BSM assumption of constant volatility. This phenomenon is a key area of research and modeling in quantitative finance (e.g., Stochastic Volatility Models).

Risk-Neutral Valuation

The BSM model is derived under the Risk-Neutral Measure ( $\mathbb{Q}$ ).

Principle: In a complete and arbitrage-free market, the price of any derivative is the discounted expected value of its future payoff, where the expectation is taken under a measure where all assets grow at the risk-free rate $r$ .
Relevance: This concept simplifies pricing by allowing us to ignore the true market risk premium and focus only on the probability distribution of the underlying asset under the risk-neutral world. The drift of the underlying asset price process is set to $r$ instead of the true expected return $\mu$ .